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Comments on “Analytical expression of explicit MPC solution via lattice piecewise-affine function” [Automatica 45 (2009) 910–917]
Automatica 48 (2012) 2993–2994
Contents lists available at SciVerse ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Correspondence
Comments on ‘‘Analytical expression of explicit MPC solution via lattice piecewise-affine function’’ [Automatica 45 (2009) 910–917]✩ Farhad Bayat 1 Department of Electrical Engineering, University of Zanjan, 45371-38111 Zanjan, Iran
article
info
Article history: Received 12 June 2011 Received in revised form 2 November 2011 Accepted 11 March 2012 Available online 13 September 2012
abstract The simplification conditions for lattice representation of the PWA control law in the explicit MPC suggested in the aforementioned article are shown, by means of a counterexample, not to be sufficient. In view of this, a modified lemma is proposed in this note for ensuring the sufficiency. © 2012 Elsevier Ltd. All rights reserved.
Keywords: Explicit model predictive control Lattice piecewise-affine function Point location Multi-parametric program Canonical representation
1. Introduction The intensive online computation required in many MPC applications can be moved offline using the so called explicit MPC (eMPC) method. The eMPC law is a piecewise-affine (PWA) function defined over a partition of polyhedral regions. When an explicit PWA controller is executed, one needs to identify which polyhedral region the measured state lies in—the so called point location—and then the corresponding affine control law is evaluated and applied. In the eMPC method the point location procedure is the most computationally complex part and restricts the maximum accessible sampling rate of closed loop system. As an alternative to the point location procedure (see e.g. Bayat, Johansen, and Jalali (2011) and references therein), an analytical expression for the PWA controller is proposed in Wen, Ma, and Ydstie (2009), by the introduction of a lattice PWA function, provided that the continuous PWA solution is available. In the aforementioned paper, in order to achieve a more compact representation of the PWA controller, the authors have proposed two row and column simplification lemmas (Lemmas 3 and 4). In this short note, it is shown that the column simplification lemma
(Lemma 4) of Wen et al. (2009) leads to misrepresentation in some cases when Lemma 4 is used iteratively. This fact is demonstrated by using a counterexample in the next section. 2. The counterexample and modification Assume a PWA function as follows (Fig. 1, solid line):
f1 = −x − 3, f2 = 1, p (x) = f3 = x + 3, f4 = −x + 3, f5 = 1,
DOI of original article: http://dx.doi.org/10.1016/j.automatica.2008.11.023. was recommended for publication in revised form by Associate Editor Richard D. Braatz under the direction of Editor André L. Tits. E-mail address: [email protected]. 1 Tel.: +98 9123412407; fax: +98 2412283204. 0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.08.009
∈ R1 ∈ R2 ∈ R3 ∈ R4 ∈ R5
= [−6 − 4] = [−4 − 2] = [−2 0] = [0 2] = [2 4].
(1)
Using the results of Wen et al. (2009), the corresponding parameter, structure and dual structure matrices are calculated as
−1 0 Φ= 1 −1
−3 1 3 3
0
✩ The material in this paper has not been presented at any conference. This paper
if x if x if x if x if x
1 0 Ψˆ = 0 0 0
1 1 Ψ = 1 1 1
1 0 1 0 0 1
0 0 1 1 1
1 1 1 1 0
1 1 1 1 1
1 1 1 0 0
0 0 0 1 1
1 1 1 1 1
0 1 0 . 0 1
(2)
Let Ii ∈ Z + denotes the index set of fi , given as Ii = {k|fk (x) ≤ fi (x), ∀x ∈ Ri }. Applying Lemma 3 to the structure matrix leads to
2994
F. Bayat / Automatica 48 (2012) 2993–2994
iteratively, one will run into trouble if the function piece fk is itself removed by another representative fs which is not a representative for fj , e.g. fs ̸= fj .
ˆ k,l = To make the result of Lemma 4 correct, a sufficient condition ψ 0, ∀l, is added in Lemma 4b which prevents the removal of all local functions with same affine functions. Lemma 4b (Modified Column Vector Simplification). Assume that P (x|Φ , Ψ ) is a lattice representation with M segments. Let Ψ = [ψij ]M ×M and Ψˆ = [ψˆ ij ]M ×M denote the primary and dual structure matrices. Then the following results hold.
(1) Given any i, j, k ∈ {1, . . . , M }, if k, j ∈ Ii and ψˆ jk = 1, then one Fig. 1. A counterexample illustrating the misrepresentation when Lemma 4 in Wen et al. (2009) is used.
the following simplified matrix in which the dash rows represent the deleted rows:
1
1
0
1
− Ψ = − 1
− −
− −
− −
1
0
1
−
−
−
−
− − . 1 −
(3)
Then using Lemma 4-(1), for {i, j, k} = {1, 2, 5}, we have {j, k} = {2, 5} ∈ Ii = I1 and ψˆ j,k = 1; thus we set ψi,j = ψ1,2 = 0. Similarly, {i, j, k} = {1, 5, 3} leads to ψ1,5 = 0 and {i, j, k} = {4, 1, 4}, {4, 2, 5} lead to ψ4,1 = 0 and ψ4,2 = 0, respectively. Then, one obtains
1 Ψ = 0
0 0
1 0
0 1
0 . 1
(4)
Since ψi,j = 0, ∀j = 2, then applying Lemma 4-(2) gives
−3 →f1 1 1 0 0 3 →f3 Ψ˜ = , 0 0 1 1 3 →f4 0 1 →f5 ˜ , Ψ˜ ) = min{max(f1 , f3 ), max(f4 , f5 )}. P (x|Φ
−1 ˜ = 1 Φ −1
(2) If ψij = 0, ∀i ∈ {1, . . . , M }, then there exist a simplified structure ˜ ∈ RM ×(M −1) such that matrix Ψ˜ ∈ R(M −1)×M and parameter matrix Φ ˜ ˜ ˜ P (x|Φ , Ψ ) = P (x|Φ , Ψ ), where Φ = [φ1 , . . . , φj−1 , φj+1 , . . . , φM ]T and Ψ˜ is the same as that defined in Lemma 3 of Wen et al. (2009). ˆ Proof. Since ψjk = 1, then fj (x) ≤ fk (x), ∀x ∈ Rj . This leads to max fj (x), fk (x) = fk (x), ∀x ∈ Rj . Therefore maxp∈Ii fp (x) =
1
ˆ kl = 0, ∀l = 1, . . . , M, without affecting the can set ψij = 0 and ψ function values of P (x|Φ , Ψ ).
maxp∈Ii ,p̸=j fp (x) , ∀x ∈ Ri . This implies that ψij = 0. This means that fj can be alternatively represented by fk in the lattice min–max form. When fj (x) = fk (x), ∀x ∈ Rj and the procedure is iterated to
ˆ kl = 0, ∀l = the next steps, even if i = i, j = k and k = s, since ψ 1, . . . , M, then ψik is not allowed to be set to 0. Note that this is a sufficient condition for resolving the misrepresentation. We emphasize that one is not required to check the equality, because when fs (x) = fk (x), then fs can be represented by fk in the following ˆ sk = 1. iterations relating to ψ Then using Lemma 4b, if {i, j, k} = {1, 2, 5}, then ψ1,2 = 0 and ˆ ψ5l = 0, ∀l = 1, . . . , 5. Similarly, {i, j, k} = {4, 1, 4}, {4, 2, 5} lead ˆ 4l = 0, ∀l = 1, . . . , 5, and ψ4,2 = 0, respectively. to ψ4,1 = 0, ψ Removing the zero column gives
(5)
It is easy to verify that (5) does not match the original PWA function for x ∈ R2 . The plot of the above lattice representation is depicted in Fig. 1 (dotted line). This issue originates from the fact that the Lemma 4 in Wen et al. (2009) has been analyzed and proved for one iteration of the simplification algorithm and a piece of the PWA function fj is removed since it can be represented by another piece fk = fj in its min–max form. When the lemma is used
1 Ψ˜ B = 0
1 0
0 1
1 , 1
˜B = Φ ˜, Φ
(6)
˜ B , Ψ˜ B ) = min{max(f1 , f3 , f5 ), max(f4 , f5 )}. P (x|Φ ˜ B , Ψ˜ B ) = p(x), ∀x ∈ [−6, 4]. It is easy to see that P (x|Φ References Bayat, F., Johansen, T. A., & Jalali, A. A. (2011). Using hash tables to manage the timestorage complexity in a point location problem: Application to explicit model predictive control. Automatica, 47(3), 571–577. Wen, C., Ma, X., & Ydstie, B. E. (2009). Analytical expression of explicit mpc solution via lattice piecewise-affine function. Automatica, 45(4), 910–917.
Contents lists available at SciVerse ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Correspondence
Comments on ‘‘Analytical expression of explicit MPC solution via lattice piecewise-affine function’’ [Automatica 45 (2009) 910–917]✩ Farhad Bayat 1 Department of Electrical Engineering, University of Zanjan, 45371-38111 Zanjan, Iran
article
info
Article history: Received 12 June 2011 Received in revised form 2 November 2011 Accepted 11 March 2012 Available online 13 September 2012
abstract The simplification conditions for lattice representation of the PWA control law in the explicit MPC suggested in the aforementioned article are shown, by means of a counterexample, not to be sufficient. In view of this, a modified lemma is proposed in this note for ensuring the sufficiency. © 2012 Elsevier Ltd. All rights reserved.
Keywords: Explicit model predictive control Lattice piecewise-affine function Point location Multi-parametric program Canonical representation
1. Introduction The intensive online computation required in many MPC applications can be moved offline using the so called explicit MPC (eMPC) method. The eMPC law is a piecewise-affine (PWA) function defined over a partition of polyhedral regions. When an explicit PWA controller is executed, one needs to identify which polyhedral region the measured state lies in—the so called point location—and then the corresponding affine control law is evaluated and applied. In the eMPC method the point location procedure is the most computationally complex part and restricts the maximum accessible sampling rate of closed loop system. As an alternative to the point location procedure (see e.g. Bayat, Johansen, and Jalali (2011) and references therein), an analytical expression for the PWA controller is proposed in Wen, Ma, and Ydstie (2009), by the introduction of a lattice PWA function, provided that the continuous PWA solution is available. In the aforementioned paper, in order to achieve a more compact representation of the PWA controller, the authors have proposed two row and column simplification lemmas (Lemmas 3 and 4). In this short note, it is shown that the column simplification lemma
(Lemma 4) of Wen et al. (2009) leads to misrepresentation in some cases when Lemma 4 is used iteratively. This fact is demonstrated by using a counterexample in the next section. 2. The counterexample and modification Assume a PWA function as follows (Fig. 1, solid line):
f1 = −x − 3, f2 = 1, p (x) = f3 = x + 3, f4 = −x + 3, f5 = 1,
DOI of original article: http://dx.doi.org/10.1016/j.automatica.2008.11.023. was recommended for publication in revised form by Associate Editor Richard D. Braatz under the direction of Editor André L. Tits. E-mail address: [email protected]. 1 Tel.: +98 9123412407; fax: +98 2412283204. 0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.08.009
∈ R1 ∈ R2 ∈ R3 ∈ R4 ∈ R5
= [−6 − 4] = [−4 − 2] = [−2 0] = [0 2] = [2 4].
(1)
Using the results of Wen et al. (2009), the corresponding parameter, structure and dual structure matrices are calculated as
−1 0 Φ= 1 −1
−3 1 3 3
0
✩ The material in this paper has not been presented at any conference. This paper
if x if x if x if x if x
1 0 Ψˆ = 0 0 0
1 1 Ψ = 1 1 1
1 0 1 0 0 1
0 0 1 1 1
1 1 1 1 0
1 1 1 1 1
1 1 1 0 0
0 0 0 1 1
1 1 1 1 1
0 1 0 . 0 1
(2)
Let Ii ∈ Z + denotes the index set of fi , given as Ii = {k|fk (x) ≤ fi (x), ∀x ∈ Ri }. Applying Lemma 3 to the structure matrix leads to
2994
F. Bayat / Automatica 48 (2012) 2993–2994
iteratively, one will run into trouble if the function piece fk is itself removed by another representative fs which is not a representative for fj , e.g. fs ̸= fj .
ˆ k,l = To make the result of Lemma 4 correct, a sufficient condition ψ 0, ∀l, is added in Lemma 4b which prevents the removal of all local functions with same affine functions. Lemma 4b (Modified Column Vector Simplification). Assume that P (x|Φ , Ψ ) is a lattice representation with M segments. Let Ψ = [ψij ]M ×M and Ψˆ = [ψˆ ij ]M ×M denote the primary and dual structure matrices. Then the following results hold.
(1) Given any i, j, k ∈ {1, . . . , M }, if k, j ∈ Ii and ψˆ jk = 1, then one Fig. 1. A counterexample illustrating the misrepresentation when Lemma 4 in Wen et al. (2009) is used.
the following simplified matrix in which the dash rows represent the deleted rows:
1
1
0
1
− Ψ = − 1
− −
− −
− −
1
0
1
−
−
−
−
− − . 1 −
(3)
Then using Lemma 4-(1), for {i, j, k} = {1, 2, 5}, we have {j, k} = {2, 5} ∈ Ii = I1 and ψˆ j,k = 1; thus we set ψi,j = ψ1,2 = 0. Similarly, {i, j, k} = {1, 5, 3} leads to ψ1,5 = 0 and {i, j, k} = {4, 1, 4}, {4, 2, 5} lead to ψ4,1 = 0 and ψ4,2 = 0, respectively. Then, one obtains
1 Ψ = 0
0 0
1 0
0 1
0 . 1
(4)
Since ψi,j = 0, ∀j = 2, then applying Lemma 4-(2) gives
−3 →f1 1 1 0 0 3 →f3 Ψ˜ = , 0 0 1 1 3 →f4 0 1 →f5 ˜ , Ψ˜ ) = min{max(f1 , f3 ), max(f4 , f5 )}. P (x|Φ
−1 ˜ = 1 Φ −1
(2) If ψij = 0, ∀i ∈ {1, . . . , M }, then there exist a simplified structure ˜ ∈ RM ×(M −1) such that matrix Ψ˜ ∈ R(M −1)×M and parameter matrix Φ ˜ ˜ ˜ P (x|Φ , Ψ ) = P (x|Φ , Ψ ), where Φ = [φ1 , . . . , φj−1 , φj+1 , . . . , φM ]T and Ψ˜ is the same as that defined in Lemma 3 of Wen et al. (2009). ˆ Proof. Since ψjk = 1, then fj (x) ≤ fk (x), ∀x ∈ Rj . This leads to max fj (x), fk (x) = fk (x), ∀x ∈ Rj . Therefore maxp∈Ii fp (x) =
1
ˆ kl = 0, ∀l = 1, . . . , M, without affecting the can set ψij = 0 and ψ function values of P (x|Φ , Ψ ).
maxp∈Ii ,p̸=j fp (x) , ∀x ∈ Ri . This implies that ψij = 0. This means that fj can be alternatively represented by fk in the lattice min–max form. When fj (x) = fk (x), ∀x ∈ Rj and the procedure is iterated to
ˆ kl = 0, ∀l = the next steps, even if i = i, j = k and k = s, since ψ 1, . . . , M, then ψik is not allowed to be set to 0. Note that this is a sufficient condition for resolving the misrepresentation. We emphasize that one is not required to check the equality, because when fs (x) = fk (x), then fs can be represented by fk in the following ˆ sk = 1. iterations relating to ψ Then using Lemma 4b, if {i, j, k} = {1, 2, 5}, then ψ1,2 = 0 and ˆ ψ5l = 0, ∀l = 1, . . . , 5. Similarly, {i, j, k} = {4, 1, 4}, {4, 2, 5} lead ˆ 4l = 0, ∀l = 1, . . . , 5, and ψ4,2 = 0, respectively. to ψ4,1 = 0, ψ Removing the zero column gives
(5)
It is easy to verify that (5) does not match the original PWA function for x ∈ R2 . The plot of the above lattice representation is depicted in Fig. 1 (dotted line). This issue originates from the fact that the Lemma 4 in Wen et al. (2009) has been analyzed and proved for one iteration of the simplification algorithm and a piece of the PWA function fj is removed since it can be represented by another piece fk = fj in its min–max form. When the lemma is used
1 Ψ˜ B = 0
1 0
0 1
1 , 1
˜B = Φ ˜, Φ
(6)
˜ B , Ψ˜ B ) = min{max(f1 , f3 , f5 ), max(f4 , f5 )}. P (x|Φ ˜ B , Ψ˜ B ) = p(x), ∀x ∈ [−6, 4]. It is easy to see that P (x|Φ References Bayat, F., Johansen, T. A., & Jalali, A. A. (2011). Using hash tables to manage the timestorage complexity in a point location problem: Application to explicit model predictive control. Automatica, 47(3), 571–577. Wen, C., Ma, X., & Ydstie, B. E. (2009). Analytical expression of explicit mpc solution via lattice piecewise-affine function. Automatica, 45(4), 910–917.